Partial sum process to check regression models with multiple correlated response: With an application for testing a change-point in profile data

• Published in: Journal of multivariate analysis Vol. 102; no. 2; pp. 281 - 291
• Main Authors: Bischoff, W., Gegg, A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.02.2011; Elsevier; Taylor & Francis LLC
• Series: Journal of Multivariate Analysis
• Subjects: Change-point problem; Combinatorics; Combinatorics. Ordered structures; Designs and configurations; Euclidean space; Exact sciences and technology; Experimental design; Hypotheses; Hypothesis testing; Linear inference, regression; Mathematical models; Mathematics; Multiple linear regression model; Multiple linear regression model Multiple residual partial sum limit process Multivariate Brownian motion Change-point problem Panel data Profile data Repeated measurements; Multiple residual partial sum limit process; Multivariate analysis; Multivariate Brownian motion; Panel data; Probability and statistics; Profile data; Regression analysis; Repeated measurements; Sciences and techniques of general use; Simulation; Statistics; Studies; 62J05; Panel data; Multiple residual partial sum limit process; Multivariate Brownian motion; 62G20; 62G10; Change-point problem; 60F05; 60G15; Multiple linear regression model; Profile data; Repeated measurements; Statistical distribution; Partial sum; Response model; Projection; Change point; Regression residual; Multivariate analysis; Statistical simulation; Linear model; Hypothesis test; Significance test; Statistical test; Least squares method; Repeated measurement; Hilbert space; Approximation theory; Linear regression; Kolmogorov Smirnov test; Multiple regression; Brownian motion; Statistical method; Statistical regression; Correlation analysis; Regression model; Experimental design; Likelihood ratio test
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2010.08.014

We consider regression models with multiple correlated responses for each design point. Under the null hypothesis, a linear regression is assumed. For the least-squares residuals of this linear regression, we establish the limit of the partial sums. This limit is a projection on a certain subspace of the reproducing Kernel Hilbert space of a multivariate Brownian motion. Based on this limit, we propose a significance test of Kolmogorov–Smirnov type to test the null hypothesis and show that this result can be used to study a change-point problem in the case of linear profile data (panel data). We compare our proposed method, which does not rely on any distributional assumptions, with the likelihood ratio test in a simulation study.